DifferentialGeometry[algebraic operations] - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : DifferentialGeometry : DifferentialGeometry/AlgebraicOperations

DifferentialGeometry[algebraic operations]

addition, subtraction, scalar multiplication, wedge product, tensor product

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

A &plus B - add two vectors, differential forms or tensors

A &minus B- subtract one vector, differential form or tensor from another

A &mult B - multiply a Maple expression by a vector, differential form or tensor

A &wedge B- form the wedge (or skew) product of a pair of differential forms or multi-vectors

A &tensor B- form the tensor product of a pair of tensors

A &algmult B - multiply two vectors in an algebra  

Parameters

A, B

-

Maple expressions, differential forms or tensors

Description

• 

In the DifferentialGeometry package the wedge product of 1-forms is defined in terms of the tensor product by αβ=αββα.

• 

When using these commands together within a single Maple expression, it is important to use parentheses to insure that the operations are executed in the correct order.

• 

In an interactive Maple session, it is usually more convenient to use the commands evalDG and DGzip to perform these basic algebraic operations.

• 

Here are the precise lists of admissible arguments for these commands.

• 

A &plus B, A &minus B -- A and B: Maple expressions, vectors, differential forms of the same degree, differential biforms of the same bidegree, tensors with the same index type and density weights. A and B must be defined on the same frame.

• 

A &mult B -- A: a Maple expression; B: a Maple expression, vector, differential form, differential biform, tensor. A and B must be defined on the same frame.

• 

A &wedge B -- A and B: Maple expressions or differential forms, differential biforms.  If A and B are forms, then the sum of their degrees cannot exceed the dimension of the frame on which they are defined. If A and B are bi-forms, then the sum of their horizontal degrees cannot exceed the dimension of the base manifold on which they are defined.  A and B must be defined on the same frame.

• 

A &tensor B -- A and B: Maple expressions, vectors, differential 1-forms, tensors.  A and B must be defined on the same frame.

• 

These commands are part of the DifferentialGeometry package, and so can be used in the forms given above only after executing the command with(DifferentialGeometry).

Examples

withDifferentialGeometry:withLieAlgebras:

 

Use DGsetup to define a three-dimensional manifold M with coordinates [x, y, z].

DGsetupx,y,z,M,verbose

The following coordinates have been protected:

x,y,z

The following vector fields have been defined and protected:

_DGvector,M,,1,1,_DGvector,M,,2,1,_DGvector,M,,3,1

The following differential 1-forms have been defined and protected:

_DGform,M,1,1,1,_DGform,M,1,2,1,_DGform,M,1,3,1

frame name: M

(1)

 

Example 1.

Create linear combinations of vector fields and differential 1-forms using &plus and &mult.

X1D_x&plusD_z

_DGvector,M,,1,1,3,1

(2)

X23z&multD_x&plus2y&multD_y

_DGvector,M,,1,3z,2,2y

(3)

X3X2&minus3z&multX1

_DGvector,M,,2,2y,3,3z

(4)

α1sinz&multdx&minuscosy&multdz

_DGform,M,1,1,sinz,3,cosy

(5)

α2cosx&multdy&pluscosz&multdz

_DGform,M,1,2,cosx,3,cosz

(6)

 

Example 2.

Create differential 2-forms using &plus and &mult and &wedge.

α32&multdx&wedgedy&plus5&multdy&wedgedz

_DGform,M,2,1,2,2,2,3,5

(7)

α4α1&wedgeα2

_DGform,M,2,1,2,sinzcosx,1,3,sinzcosz,2,3,cosycosx

(8)

α5α1&wedgeα2&minusα3

_DGform,M,2,1,2,2+sinzcosx,1,3,sinzcosz,2,3,5+cosycosx

(9)

α6α1&wedgeα3

_DGform,M,3,1,2,3,2cosy+5sinz

(10)

 

Example 3.

Create various tensors using &plus, &mult and &tensor.

T1X1&tensorX1

_DGtensor,M,con_bas,con_bas,,1,1,1,1,3,1,3,1,1,3,3,1

(11)

T2X1&tensorα1

_DGtensor,M,con_bas,cov_bas,,1,1,sinz,1,3,cosy,3,1,sinz,3,3,cosy

(12)

T31&tensordx&wedgedy

_DGtensor,M,cov_bas,cov_bas,,1,2,1,2,1,1

(13)

T4dx&tensordx&tensorD_y&tensorD_z&tensordz

_DGtensor,M,cov_bas,cov_bas,con_bas,con_bas,cov_bas,,1,1,2,3,3,1

(14)

T51y2&multdx&tdx+dy&tdy

_DGtensor,M,cov_bas,cov_bas,,1,1,1y2,2,2,1y2

(15)

 

Example 4.

Create a multi-vector using &plus, &mult and &tensor.

V12&multD_x&wedgeD_y&plus3&multD_y&wedgeD_z

_DGmultivector,M,2,1,2,2,2,3,3

(16)

Example 5.

Use the command AlgebraLibraryData to retrieve the structure equations for the quaternions.

LAAlgebraLibraryDataQuaternions,Q

LAe12=e1,e1·e2=e2,e1·e3=e3,e1·e4=e4,e2·e1=e2,e22=e1,e2·e3=e4,e2·e4=e3,e3·e1=e3,e3·e2=e4,e32=e1,e3·e4=e2,e4·e1=e4,e4·e2=e3,e4·e3=e2,e42=e1

(17)

Initialize.

DGsetupLA,e,i,j,k,θ

algebra name: Q

(18)

Calculate some simple sums and products of quaternions.

Q1i&algmultj

_DGvector,Q,,4,1

(19)

Q2e&plusi&plusj&plusk&algmulte&minusi&plusj&plusk

_DGvector,Q,,1,4

(20)
Q > 

See Also

DifferentialGeometry

DGzip

evalDG